Chapter 8 Statistical inference: Significance Tests About Hypotheses

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Chapter 8Statistical inference Significance
Tests About Hypotheses

• Learn .
• To use an inferential method called
• a Significance Test
• To analyze evidence that data provide
• To make decisions based on data

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Two Major Methods for Making Statistical
Inferences about a Population

• Confidence Interval
• Significance Test

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Questions that Significance Tests Attempt to
Answer

• Does a proposed diet truly result in weight loss,
  on the average?
• Is there evidence of discrimination against women
  in promotion decisions?
• Does one advertising method result in better
  sales, on the average, than another advertising
  method?

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Section 8.1

• What Are the Steps For Performing a Significance
  Test?

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Hypothesis

• A hypothesis is a statement about a population,
  usually of the form that a certain parameter
  takes a particular numerical value or falls in a
  certain range of values
• The main goal in many research studies is to
  check whether the data support certain hypotheses

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Significance Test

• A significance test is a method of using data to
  summarize the evidence about a hypothesis
• A significance test about a hypothesis has five
  steps

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Step 1 Assumptions

• A (significance) test assumes that the data
  production used randomization
• Other assumptions may include
• Assumptions about the sample size
• Assumptions about the shape of the population
  distribution

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Step 2 Hypotheses

• Each significance test has two hypotheses
• The null hypothesis is a statement that the
  parameter takes a particular value
• The alternative hypothesis states that the
  parameter falls in some alternative range of
  values

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Null and Alternative Hypotheses

• The value in the null hypothesis usually
  represents no effect
• The symbol Ho denotes null hypothesis
• The value in the alternative hypothesis usually
  represents an effect of some type
• The symbol Ha denotes alternative hypothesis

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Null and Alternative Hypotheses

• A null hypothesis has a single parameter value,
  such as Ho p 1/3
• An alternative hypothesis has a range of values
  that are alternatives to the one in Ho such as
• Ha p ? 1/3 or
• Ha p gt 1/3 or
• Ha p lt 1/3

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Step 3 Test Statistic

• The parameter to which the hypotheses refer has a
  point estimate the sample statistic
• A test statistic describes how far that estimate
  (the sample statistic) falls from the parameter
  value given in the null hypothesis

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Step 4 P-value

• To interpret a test statistic value, we use a
  probability summary of the evidence against the
  null hypothesis, Ho
• First, we presume that Ho is true
• Next, we consider the sampling distribution from
  which the test statistic comes
• We summarize how far out in the tail of this
  sampling distribution the test statistic falls

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Step 4 P-value

• We summarize how far out in the tail the test
  statistic falls by the tail probability of that
  value and values even more extreme
• This probability is called a P-value
• The smaller the P-value, the stronger the
  evidence is against Ho

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Step 4 P-value
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Step 4 P-value

• The P-value is the probability that the test
  statistic equals the observed value or a value
  even more extreme
• It is calculated by presuming that the null
  hypothesis H is true

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Step 5 Conclusion

• The conclusion of a significance test reports the
  P-value and interprets what it says about the
  question that motivated the test

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Summary The Five Steps of a Significance Test

• Assumptions
• Hypotheses
• Test Statistic
• P-value
• Conclusion

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Is the Statement a Null Hypothesis or an
Alternative Hypothesis?

• In Canada, the proportion of adults who favor
  legalize gambling is 0.50.
• Null Hypothesis
• Alternative Hypothesis

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Is the Statement a Null Hypothesis or an
Alternative Hypothesis?

• The proportion of all Canadian college students
  who are regular smokers is less than 0.24, the
  value it was ten years ago.
• Null Hypothesis
• Alternative Hypothesis

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Section 8.4

• Decisions and Types of Errors in Significance
  Tests

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Type I and Type II Errors

• When H0 is true, a Type I Error occurs when H0 is
  rejected
• When H0 is false, a Type II Error occurs when H0
  is not rejected

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Significance Test Results
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An Analogy Decision Errors in a Legal Trial
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P(Type I Error) Significance Level a

• Suppose H0 is true. The probability of rejecting
  H0, thereby committing a Type I error, equals the
  significance level, a, for the test.

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P(Type I Error)

• We can control the probability of a Type I error
  by our choice of the significance level
• The more serious the consequences of a Type I
  error, the smaller a should be

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Type I and Type II Errors

• As P(Type I Error) goes Down, P(Type II Error)
  goes Up
• The two probabilities are inversely related

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A significance test about a proportion is
conducted using a significance level of 0.05.

• The test statistic is 2.58. The P-value is 0.01.
  If Ho is true, for what probability of a Type I
  error was the test designed?
• .01
• .05
• 2.58
• .02

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A significance test about a proportion is
conducted using a significance level of 0.05.

• The test statistic is 2.58. The P-value is 0.01.
  If this test resulted in a decision error, what
  type of error was it?
• Type I
• Type II